### E = m c^{2}

## Energy and Inertia

*unifying mass and energy*

### introduction

Relativity is a great unifying theory. Many fundamental concepts that were considered to be independent, prior to relativity, were brought together under one concept. We have developed all the necessary relationships in the course so far to derive the most important unification to come out of relativity. The unification of mass and energy.

I shall use the results of the last section, The Energy of Beamed Photons, to present a similar derivation to that presented by Einstein in his original paper. I shall not use mathematics, but make it a little sweater.

### consider a cake

Anyone on a diet knows that a cake has energy. That energy can be measured in Joules. When digested you may be able to extract 30 million Joules of energy from the cake. Well over your daily requirements of about 9 million Joules. You could expend approximately 10 Joules of that energy by lifting 10kg vertically 10cm. Any energy you did not use may then be stored as fat by your body. In this section a cake is going to feature and it will represent a bundle of energy.

Instead of a spacecraft flying through space I want you to consider a cake flying through space. Now since energy can transform, from one type of energy to another, I want you to imagine that there is a way to extract the energy in the cake to make light. On the cake are two minute light emitting diodes. One emits a photon directly above the cake and the other emits an identical photon in the opposite direction, directly below the cake.

Now when considering the cake I want you to think of it just in terms of its energy content. Lets call this energy the "internal energy" of the cake. If the cake emits two photons, then the internal energy of the cake is reduced by the total energy of the two photons. There is now less energy available from the cake.

Another way to reduce the energy available from the cake would be to cut a slice from the cake and throw it away. Once again there is less energy available from the cake.

Now consider a specific example. We wish to decrease the energy of the whole cake by one quarter. We could

- cut a slice equivalent to a quarter of the cake and throw it away.
- emit two identical very energetic photons that in total carry a quarter of the energy away.

In either case if the extraction of energy is done carefully the cake will remain stationary during the whole process. In both instances if either case is repeated 4 times then there would be no more internal energy available.

### consider a cake in motion

If our cake is in motion, besides the internal energy, it will now also possess the energy of motion. This is called kinetic energy. The faster it moves the more kinetic energy it possess. In the 17^{th} century this energy was found to be proportional to the square of the velocity. For example, doubling the speed of the cake would give it four times the kinetic energy.

For a given velocity the kinetic energy is also proportional to the mass of the cake. Obviously two identical cakes moving along together possess twice the kinetic energy of a single cake. Conversely, reduce the mass by a half and the kinetic energy is halved.

Now referring back to the photons that carry a quarter of the internal energy away from the cake when the cake is stationary. From the previous section, The Energy of Beamed Photons, the faster the cake moves the more energy the photons take away from the cake. If the photons carry a quarter of the energy of the cake away when the cake is stationary, they still take a quarter of the total energy of the cake away when in motion. The total energy content of the cake is the internal energy plus the kinetic energy. The photons take away a quarter of the internal energy and a quarter of the kinetic energy.

Consider a situation with the cake stationary and we continue to emit photons until all the internal energy is used up. If each pair of photons emit one quarter of the internal energy then 4 pairs of photons would emit all the internal energy. In this situation there is no more energy to create photons. If the same situation is observed from a moving reference frame the cake is seen to be moving. The cake has more energy, internal energy plus kinetic energy. A pair of photons must take away a quarter of the energy so that 4 pairs takes away all the energy. Just because the cake is moving in one reference frame it does not mean it is allowed to emit more photons than in the stationary frame. The kinetic energy of the cake must be divided evenly between the 4 pairs of photons and so the photon energy increases by the square of the velocity of the cake just like the kinetic energy.

### reducing the kinetic energy of the cake

Now consider some methods for reducing the kinetic energy of the cake. I want to reduce the energy by one quarter. Knowing that kinetic energy is dependant on mass and velocity the following are two possible ways of reducing the kinetic energy of the cake.

- reduce the velocity by colliding the cake with another object.
- reduce the mass by carefully taking a section of the cake away without changing its velocity.

In these two instances the kinetic energy of the cake is reduced. In the first instance the velocity of the cake is reduced in the second the mass, or inertia, of the cake is reduced.

Another way to reduce the kinetic energy of the cake is to emit two photons, as discussed above. If the photons carry away one quarter of the internal energy of a stationary cake, then three quarters of its original energy is left. It is extremely important to note that while the cake is stationary it does not move during the process of emitting the two photons. The cake is stationary before and after the emission of the photons. It can then be said that if the same process is observed with the cake moving then the velocity of the cake does not change during the emission of the photons.

Here is the important conclusion. From the statements above,

**The only way to reduce the kinetic energy of the cake without reducing its velocity is to take mass away. Since the photons are taking some kinetic energy away without changing the velocity of the cake they must also be taking away some mass, or inertia.**

In our example the photons take a quarter of the total energy away. In the process it is decreasing the kinetic energy of the cake by one quarter. To do this without changing the velocity of the cake they must be taking a quarter of the mass away. Emit four pairs of photons , all the energy of the cake will be taken away along with all the mass. There would be no cake left.

The direct implication is the energy of the photon carries with it inertia. And so inertia, or mass, is taken away from the cake.

If we give the cake more kinetic energy the photons will emit more energy and so take away more mass. It would seem that both photons and kinetic energy have inertia. Einstein generalized this relationship for all forms of energy. All energy behaves as though it has inertia and all inertia has energy.

### conclusion

Photons have provided us with a pure form of energy to develop this concept that photons have inertia. The argument presented so far shows that that energy has inertia. As energy is taken away from the cake so is inertia. Fundamentally we measure the mass of an object through this property of inertia.

Einstein produced a paper in September of 1905 entitled "Does the Inertia of a Body depend upon its Energy-Content?". In the conclusion of the paper he answered this question by writing the relationship between inertia and energy. Today the symbols used to express this relationship are different but the meaning is the same.